Chapter 2 Equations and Inequalities
2.3 Complex Numbers
Learning Objectives
In this section, you will:
- Add and subtract complex numbers.
- Multiply complex numbers.
- Divide complex numbers.
- Simplify powers of [latex]i[/latex].
Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple.
In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. In turn, the set of rational numbers fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it.
Expressing Square Roots of Negative Numbers as Multiples of [latex]\,i[/latex]
We know how to find the square root of any positive real number. Similarly, we can find the square root of any negative number. The difference is that the root is not real. If the number inside the radicand is negative, the root is said to be an imaginary number. The imaginary number[latex]\,i\,[/latex]is defined as the square root of[latex]\,-1.[/latex]
So, using the properties of radicals,
[latex]{i}^{2}={\left(\sqrt{-1}\right)}^{2}=-1[/latex]
We can write the square root of any negative number as a multiple of[latex]\,i.\,[/latex]Consider the square root of [latex]\,-49.[/latex]
[latex]\begin{array}{ccc}\hfill \sqrt{-49}& =& \hfill \sqrt{49\cdot \left(-1\right)}\\ & =& \sqrt{49}\sqrt{-1}\hfill \\ & =& 7i\hfill \end{array}[/latex]
We use[latex]\,7i\,[/latex]and not[latex]\,-7i\,[/latex]because the principal root of[latex]\,49\,[/latex]is the positive root.
A complex number is the sum of real and imaginary numbers. A complex number is expressed in standard form when written[latex]\,a+bi\,[/latex]where[latex]\,a\,[/latex]is the real part and[latex]\,b\,[/latex]is the imaginary part. For example,[latex]\,5+2i\,[/latex]is a complex number. So, too, is[latex]\,3+4i\sqrt{3}.[/latex]
Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number, and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.
Complex Numbers
A complex number is a number of the form[latex]\,a+bi,[/latex] where
- [latex]\,i=\sqrt{-1}[/latex]
- [latex]a\,[/latex]is the